\section*{Part C}
The lifetime (expressed in hours) of the electronic panel displaying the competitors' scores is a random variable $T$ that follows the exponential distribution with parameter $\lambda = 10^{-4}$ (expressed in $\mathrm{h}^{-1}$).

\begin{enumerate}
  \item What is the probability that the panel functions for at least 2000 hours?
  \item Organized presentation of knowledge
\end{enumerate}

In this question, $\lambda$ denotes a strictly positive real number.\\
Recall that the mathematical expectation of the random variable $T$ following an exponential distribution with parameter $\lambda$ is defined by: $\mathrm{E}(T) = \lim_{x \rightarrow +\infty} \int_{0}^{x} \lambda t \mathrm{e}^{-\lambda t} \mathrm{~d}t$.\\
a. Consider the function $F$, defined for all real $t$ by: $F(t) = \left(-t - \frac{1}{\lambda}\right) \mathrm{e}^{-\lambda t}$.

Prove that the function $F$ is an antiderivative on $\mathbb{R}$ of the function $f$ defined for all real $t$ by: $f(t) = \lambda t \mathrm{e}^{-\lambda t}$.\\
b. Deduce that the mathematical expectation of the random variable $T$ is equal to $\frac{1}{\lambda}$.

What is the expected lifetime of the electronic panel displaying the competitors' scores?